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My professor gives a multiclass confusion matrix and asks for the error rate of a certain class. Unfortunately, the professor refuses to give a definition.

I think the closest value to an error rate for a class $j$ is the conditional probability $\mathrm{P}(\mathrm{Pred} \neq j ~|~ \mathrm{Truth} = j)$, i.e. the sum of the offdiagonal entries along the $j$-column divided by the sum of all the $j$-column entries. Do you agree?

(I guess the only alternative would be $\mathrm{P}(\mathrm{Truth} \neq j ~|~ \mathrm{Pred} = j)$, which is computed along the $j$-row.)

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I don't know if this is the definition that you're supposed to use, but usually in multiclass classification the most standard method to apply a binary classification measure is one-vs-rest: given a target class $C$ (positive class), consider all the other classes as a single negative class (i.e. as if they are all merged together).

According to this interpretation, the error rate of a target class $j$ would be the probability of an instance to have different predicted and true class, excluding cases where neither the predicted or true class is $j$ (these are not counted as errors since both the predicted and true class are "negative"),

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  • $\begingroup$ Many thanks, @Erwan! Correct me if I am wrong: this is then the sum of all the entries along the $j$-row and $j$-column, excluding the diagonal entry, divided by the sum of the $j$-row and $j$-column, including the diagonal entry? $\endgroup$
    – Pippo
    Jun 14 at 16:37
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    $\begingroup$ @Pippo: correct for the numerator, but the denominator should be the full sum of the matrix (i.e. total number of instances). $\endgroup$
    – Erwan
    Jun 14 at 19:20

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